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Theorem cvlatcvr1 30213
Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlatcvr1.j  |-  .\/  =  ( join `  K )
cvlatcvr1.c  |-  C  =  (  <o  `  K )
cvlatcvr1.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlatcvr1  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C
( P  .\/  Q
) ) )

Proof of Theorem cvlatcvr1
StepHypRef Expression
1 simp13 990 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  CvLat )
2 cvlatl 30197 . . . 4  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
31, 2syl 16 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  K  e.  AtLat )
4 eqid 2438 . . . 4  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2438 . . . 4  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 cvlatcvr1.a . . . 4  |-  A  =  ( Atoms `  K )
74, 5, 6atnem0 30190 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  ( P
( meet `  K ) Q )  =  ( 0. `  K ) ) )
83, 7syld3an1 1231 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  ( P
( meet `  K ) Q )  =  ( 0. `  K ) ) )
9 eqid 2438 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
109, 6atbase 30161 . . 3  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
11 cvlatcvr1.j . . . 4  |-  .\/  =  ( join `  K )
12 cvlatcvr1.c . . . 4  |-  C  =  (  <o  `  K )
139, 11, 4, 5, 12, 6cvlcvrp 30212 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  ( Base `  K
)  /\  Q  e.  A )  ->  (
( P ( meet `  K ) Q )  =  ( 0. `  K )  <->  P C
( P  .\/  Q
) ) )
1410, 13syl3an2 1219 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  (
( P ( meet `  K ) Q )  =  ( 0. `  K )  <->  P C
( P  .\/  Q
) ) )
158, 14bitrd 246 1  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  =/=  Q  <->  P C
( P  .\/  Q
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   joincjn 14406   meetcmee 14407   0.cp0 14471   CLatccla 14541   OMLcoml 30047    <o ccvr 30134   Atomscatm 30135   AtLatcal 30136   CvLatclc 30137
This theorem is referenced by:  cvlatcvr2  30214  atcvr1  30288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194
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